已知函数f(x)=2sinxcosx+cos2x(x属于R)

题目

已知函数f(x)=2sinxcosx+cos2x(x属于R)
1.求f(x)的最小正周期和最大值
2.若θ为锐角,且f(θ+π/8)=3分之根号2,求tan2θ的值

答案

f(x)=sin2x+cos2x
=√2sin(2x+π/4)
所以T=2π/2=π
最大值=√2
f(θ+π/8)=√2sin(2θ+π/4+π/4)
=√2cos2θ
=√2/3
cos2θ=1/3
θ锐角则sin2θ>0
sin²2θ+cos²2θ=1
sin2θ=2√2/3
tan2θ=2√2